Ultrathin Al2O3 Films Research Articles

Ultrathin Al2O3 film modification on waterborne epoxy coatings by atomic layer deposition for augmenting the corrosion resistance

The polar channels formed by the curing of waterborne anticorrosive coatings compromise their water resistance, leading to coating degradation and metal corrosion. To enhance the anticorrosive performance of waterborne coatings, this study proposed a novel method of depositing ultrathin Al2O3 films on the surface of waterborne epoxy coatings by atomic layer deposition, a technique that can modify the surface properties of polymer materials by depositing functional films. The Al2O3-modified coatings exhibited improved sealing and barrier properties by closing the polar channels and surface defects and cracks. The surface structure and morphology of the modified coatings were characterized by x-ray photoelectron spectroscopy and scanning electron microscopy. The hydrophilicity and corrosion resistance of the modified coatings were evaluated by water contact angle measurement, Tafel polarization curve, and electrochemical impedance spectroscopy. The results indicated that the water contact angle of the Al2O3-modified coating increased by 48° compared to the unmodified coating, and the protection efficiency of the modified coating reached 99.81%. The Al2O3-modified coating demonstrated high anticorrosive efficiency and potential applications for metal anticorrosion in harsh marine environments.

Combination of Multiple Operando and In-Situ Characterization Techniques in a Single Cluster System for Atomic Layer Deposition: Unraveling the Early Stages of Growth of Ultrathin Al2O3 Films on Metallic Ti Substrates

This work presents a new ultra-high vacuum cluster tool to perform systematic studies of the early growth stages of atomic layer deposited (ALD) ultrathin films following a surface science approach. By combining operando (spectroscopic ellipsometry and quadrupole mass spectrometry) and in situ (X-ray photoelectron spectroscopy) characterization techniques, the cluster allows us to follow the evolution of substrate, film, and reaction intermediates as a function of the total number of ALD cycles, as well as perform a constant diagnosis and evaluation of the ALD process, detecting possible malfunctions that could affect the growth, reproducibility, and conclusions derived from data analysis. The homemade ALD reactor allows the use of multiple precursors and oxidants and its operation under pump and flow-type modes. To illustrate our experimental approach, we revisit the well-known thermal ALD growth of Al2O3 using trimethylaluminum and water. We deeply discuss the role of the metallic Ti thin film substrate at room temperature and 200 °C, highlighting the differences between the heterodeposition (<10 cycles) and the homodeposition (>10 cycles) growth regimes at both conditions. This surface science approach will benefit our understanding of the ALD process, paving the way toward more efficient and controllable manufacturing processes.

Three-Dimensional Surface Treatment of MoS2 Using BCl3 Plasma-Derived Radicals.

The realization of next-generation gate-all-around field-effect transistors (FETs) using two-dimensional transition metal dichalcogenide (TMDC) semiconductors necessitates the exploration of a three-dimensional (3D) and damage-free surface treatment method to achieve uniform atomic layer-deposition (ALD) of a high-k dielectric film on the inert surface of a TMDC channel. This study developed a BCl3 plasma-derived radical treatment for MoS2 to functionalize MoS2 surfaces for the subsequent ALD of an ultrathin Al2O3 film. Microstructural verification demonstrated a complete coverage of an approximately 2 nm-thick Al2O3 film on a planar MoS2 surface, and the applicability of the technique to 3D structures was confirmed using a suspended MoS2 channel floating from the substrate. Density functional theory calculations supported by optical emission spectroscopy and X-ray photoelectron spectroscopy measurements revealed that BCl radicals, predominantly generated by the BCl3 plasma, adsorbed on MoS2 and facilitated the uniform nucleation of ultrathin ALD-Al2O3 films. Raman and photoluminescence measurements of monolayer MoS2 and electrical measurements of a bottom-gated FET confirmed negligible damage caused by the BCl3 plasma-derived radical treatment. Finally, the successful operation of a top-gated FET with an ultrathin ALD-Al2O3 (∼5 nm) gate dielectric film was demonstrated, indicating the effectiveness of the pretreatment.

Low temperature, highly stable ZnO thin-film transistors

A low-temperature and straightforward fabrication process for ZnO thin-film transistors (TFTs) with near-zero aging and negligible instability enabled by using an ultrathin oxide as a top-passivation layer is demonstrated. The process features bottom-gate top-contacts ZnO TFTs with ultrathin HfO2 or Al2O3 as passivation layers on top of the TFT followed by post-fabrication annealing (PFA). Devices with ultra-thin capping films of Al2O3 followed by a 150 °C PFA show threshold voltage shift (ΔVTH) of <1% after bias stress and negligible shift after aging. The devices show saturation threshold voltage (VTH-SAT) of 2.70 V, saturation mobilities larger than 10 cm2/V·s, and current ION/IOFF ratios >106. On the contrary, devices without nanofilm show similar performance to those with Al2O3 but show more considerable instability to aging and bias stress (ΔVTH > 5%). Also, devices with HfO2 as a capping layer shows severe instability (ΔVTH > 40%). A degradation mechanism to explain the improved aging and reliability performance is also discussed.

Direct atomic layer deposition of ultra-thin Al2O3 and HfO2 films on gold-supported monolayer MoS2

In this paper, the atomic layer deposition (ALD) of ultra-thin films (<4 nm) of Al2O3 and HfO2 on gold-supported monolayer (1L) MoS2 is investigated, providing an insight on the mechanisms ruling the nucleation in the early stages of the ALD process. A preliminary multiscale characterization of large area 1L-MoS2 exfoliated on sputter-grown Au/Ni films demonstrated: (i) a tensile strain (from 0.1 to 0.3%) and p-type doping (from 1 × 1012 to 4 × 1012 cm−2) distribution at micro-scale; (ii) an almost conformal MoS2 membrane to the Au grains topography, with some locally detached regions, indicating the occurrence of strain variations at the nanoscale; (iii) atomic scale variability (from ∼ 4.0 to ∼ 4.5 Å) in the Mo-Au atomic distances was detected, depending on the local configuration of Au nanograins. Ab initio DFT calculations of a free-standing MoS2 layer and a simplified MoS2/Au(111) interface model showed a significant influence of the Au substrate on the MoS2 energy band structure, whereas small differences were accounted for the adsorption of H2O, TMA (co-reactant, and Al-precursor, respectively) molecules, and a slight improved adsorption was predicted for TDMAHf (Hf-precursor). This suggests a crucial role of nanoscale morphological effects, such as the experimentally observed local curvature and strain of the MoS2 membrane, in the enhanced physisorption of the precursors. Afterwards the nucleation and growth of Al2O3 an HfO2 films onto 1L-MoS2/Au was investigated in detail, by monitoring the surface coverage as a function of the number (N) of ALD cycles, with N from 10 to 120. At low N values, a slower growth rate of the initially formed nuclei was observed for HfO2, probably due to the bulky nature of the TDMAHf precursor as compared to TMA. On the other hand, the formation of continuous films was obtained in both cases for N > 80 ALD cycles, corresponding to ∼ 3.6 nm Al2O3 and ∼ 3.1 nm HfO2. Current mapping on these ultra-thin films by conductive-AFM showed, for the same applied bias, a uniform insulating behavior of Al2O3 and the occurrence of few localized breakdown spots in the case of HfO2, associated to a less compact films regions. Finally, an increase of the 1L-MoS2 tensile strain was observed by Raman mapping after encapsulation with both high-κ films, accompanied by a reduction in the PL intensity, explained by the effects of strain and the higher effective dielectric constant of the surrounding environment.

Synthesis of TiO2/Al2O3 Double-Layer Inverse Opal by Thermal and Plasma-Assisted Atomic Layer Deposition for Photocatalytic Applications.

In comparison to conventional nano-infiltration approaches, the atomic layer deposition (ALD) technology exhibits greater potential in the fabrication of inverse opals (IOs) for photocatalysts. In this study, TiO2 IO and ultra-thin films of Al2O3 on IO were successfully deposited using thermal or plasma-assisted ALD and vertical layer deposition from a polystyrene (PS) opal template. SEM/EDX, XRD, Raman, TG/DTG/DTA-MS, PL spectroscopy, and UV Vis spectroscopy were used for the characterization of the nanocomposites. The results showed that the highly ordered opal crystal microstructure had a face-centered cubic (FCC) orientation. The proposed annealing temperature efficiently removed the template, leaving the anatase phase IO, which provided a small contraction in the spheres. In comparison to TiO2/Al2O3 plasma ALD, TiO2/Al2O3 thermal ALD has a better interfacial charge interaction of photoexcited electron-hole pairs in the valence band hole to restrain recombination, resulting in a broad spectrum with a peak in the green region. This was demonstrated by PL. Strong absorption bands were also found in the UV regions, including increased absorption due to slow photons and a narrow optical band gap in the visible region. The results from the photocatalytic activity of the samples show decolorization rates of 35.4%, 24.7%, and 14.8%, for TiO2, TiO2/Al2O3 thermal, and TiO2/Al2O3 plasma IO ALD samples, respectively. Our results showed that ultra-thin amorphous ALD-grown Al2O3 layers have considerable photocatalytic activity. The Al2O3 thin film grown by thermal ALD has a more ordered structure compared to the one prepared by plasma ALD, which explains its higher photocatalytic activity. The declined photocatalytic activity of the combined layers was observed due to the reduced electron tunneling effect resulting from the thinness of Al2O3.

Growth of ultrathin Al2O3 films on Polydopamine-modified polyethylene terephthalate by atomic layer deposition

Atomic layer deposition (ALD) has drawn considerable attention recently for depositing functional films on polymers. However, most polymers are inert and thermally fragile, thus posing many challenges during deposition. This study presents a novel strategy for ALD of an ultrathin Al2O3 film on alkali-treated polyethylene terephthalate (PET) via polydopamine (PDA) modification. The tightly adherent PDA layer exhibited numerous robust active sites (phenolic hydroxyl groups), which are highly desirable for precursor chemisorption during ALD. The modification process was systematically characterized. The PET substrate showed many active sites and a smooth surface after 24 h of PDA deposition. The island growth mode was found during the ALD of an Al2O3 film on the samples. The PDA layer absorbed a large quantity of the precursor (trimethylaluminum [TMA]) during the initial deposition, contributing to the early nucleation and continuous growth of the Al2O3 film. In addition, the presence of the PDA layer narrowed the region of TMA infiltration into the PET, resulting in the formation of a conformal film on the surface. Transmission electron microscopy indicated that an Al2O3 film with a thickness of about 10 nm was formed on the PDA-modified PET substrate after 100 ALD cycles, and exhibited excellent water vapor blocking characteristics. These findings suggest that the added PDA layer has tremendous potential to act as a bonding layer between polymers and ALD functional films.

Enhancement in electrical and optical properties of field-effect passivated GaN blue light emitting diodes

This paper investigates the field-effect sidewall surface passivation of III-Nitride blue light emitting diodes (LEDs) using ultrathin films of Al2O3 and ZnO grown using atomic layer deposition. A large density of negative fixed charges existing at the sidewall surface of the passivation layer leads to formation of an electric field at the interface of the passivation layer and the GaN layer. This repels the free negative charge carriers from getting close to the surface of the semiconductor and being trapped there. It also reduces the non-radiative recombination of these charge carriers and eventually leads to enhancement in electrical and optical properties of these devices. Electrical measurements demonstrate a significant decrease in the leakage current and the ideality factor in addition to a notable increase in the parasitic shunt resistance of the field-effect-passivated GaN LEDs compared to non-passivated devices. The field-effect passivation can be especially useful for smaller LED die size. In such a device, the effect of high leakage current at the edges of the device is more detrimental to its performance. The proposed approach can further eliminate the shunt conduction paths along the sidewalls of the LED and increase the efficiency of the devices.

Direct Atomic Layer Deposition of Ultrathin Aluminum Oxide on Monolayer MoS2 Exfoliated on Gold: The Role of the Substrate

AbstractIn this paper, the authors demonstrate the atomic layer deposition (ALD) of highly homogeneous and ultrathin (≈3.6 nm) Al2O3 films with very good insulating properties (breakdown field of ≈10–12 MV cm−1) directly onto monolayer (1L) MoS2 exfoliated on gold. Differently than in the case of 1L MoS2 supported by a common insulating substrate (Al2O3/Si), a better nucleation process of the high‐k film is observed on the 1L MoS2/Au system since the ALD early stages. Atomic force microscopy analyses show a ≈50% Al2O3 surface coverage just after 10 ALD cycles, its increase to &gt;90% (after 40 cycles), and a uniform ≈3.6 nm film (after 80 cycles). The Al2O3 density on bilayer MoS2 is found to be significantly reduced with respect to 1L MoS2/Au, suggesting a role of screened interface charges with the metal substrate on the adsorption of ALD precursors. Finally, Raman and photoluminescence spectroscopy show a p‐type doping and tensile strain of 1L MoS2 induced by the Au substrate, providing an insight on the evolution of vibrational and optical properties after the Al2O3 deposition. The direct ALD growth of Al2O3 on large‐area 1L MoS2 induced by the Au underlayer can be of wide interest for electronic applications.

Long-life LiNi0.5Mn1.5O4/graphite lithium-ion cells with an artificial graphite-electrolyte interface

The high operating voltage of a spinel LiNi0.5Mn1.5O4 (LNMO) cathode leads to electrolyte decomposition and accelerated deterioration of the electrode/electrolyte interface in lithium-ion cells. Aggressive side reactions prevent long-term cycling and hinder its practical application. Advanced characterization shows that the high-voltage operation makes the graphite anode react with the electrolyte, particularly with the lithium salt in the electrolyte, resulting in massive active lithium inventory loss and capacity fade. The side reactions at the graphite are identified as the main culprit for the LNMO/graphite full cell decay. Here, we demonstrate that an artificial solid-electrolyte interface layer consisting of an ultra-thin Al2O3 film, generated via atomic layer deposition (ALD) at the graphite surface, can successfully isolate the graphite from the electrolyte and inhibit the undesired reactions. Consequently, the capacity and cycling stability of the LNMO/graphite cell are drastically improved. After 300 cycles, the capacity retention of LNMO/graphite cell increases from 57.7% to 98.6% with the ALD-coated graphite. Our findings provide valuable insights into the LNMO/graphite cell degradation and suggest a simple and robust, yet highly effective, approach for the practical viability of high-voltage lithium-ion batteries.

Aluminum oxide nucleation in the early stages of atomic layer deposition on epitaxial graphene

The nucleation and growth mechanism of aluminum oxide (Al2O3) in the early stages of atomic layer deposition (ALD) on monolayer epitaxial graphene (EG) on silicon carbide (4H–SiC) has been investigated by atomic force microscopy (AFM), conductive-atomic force microscopy (C-AFM) and Raman spectroscopy. Differently than for other types of graphene, a large and uniform density of nucleation sites was observed in the case of EG and ascribed to the presence of the buffer layer at EG/SiC interface. The deposition process was characterized by Al2O3 island growth in the very early stages, followed by the formation of a continuous Al2O3 film (∼2.4 nm thick) after only 40 ALD cycles due to the islands coalescence, and subsequent layer-by-layer growth. The electrical insulating properties of the deposited ultrathin Al2O3 films were demonstrated by nanoscale current mapping with C-AFM. Raman spectroscopy analyses showed low impact of the ALD process on the defect’s density of EG. The EG strain was also almost unaffected by the deposition in the regime of island growth and coalescence, whereas a significant increase was observed after the formation of a compact Al2O3 film. The obtained results can have important implications for device applications of epitaxial graphene requiring ultra-thin high-k insulators.

Electron transport mechanism through ultrathin Al2O3 films grown at low temperatures using atomic–layer deposition

Alumina (Al2O3) films of different thicknesses have been grown at different low temperatures (100 °C–250 °C) by atomic–layer deposition on n–type Si substrate. The robustness of the Al2O3 film as a barrier has been investigated based on Al/Al2O3/Si metal–insulator–semiconductor structures. The electron transport through the Al2O3 layer was fitted well by the Fowler–Nordheim tunneling mechanism, from which the barrier heights (conduction band offset between Si and Al2O3) were deduced. It was discovered that the growth temperature and film thickness both influenced the carrier transport and barrier height. The Al/Al2O3/Si structure with an ultrathin 3 nm Al2O3 fabricated at 150 °C showed the largest barrier height, the lowest tunneling current density (4.9 × 10–8 A cm−2 at 5 MV cm−1), and the highest breakdown field strength of 18.3 MV cm−1. Using Au to replace Al as the electrode could suppress the tunneling current significantly. The Al2O3 films were also examined by x-ray photoelectron spectroscopy to determine their chemical constituents.

Growth of Ultrathin Al2O3 Films on n-InP Substrates as Insulating Layers by RF Magnetron Sputtering and Study on the Optical and Dielectric Properties

Here, we report an explorative study of an attempt to fabricate ultrathin aluminum oxide films on n-InP substrates by radio-frequency (RF) magnetron sputtering as a candidate for insulating layers in semiconductor lasers for optical communication. Film thickness and morphology were monitored to study the film growth and to explore the minimum thickness of a continuous film that RF magnetron sputtering could achieve. Originating from the weak wettability between the n-InP substrate and the Al2O3 film, Al2O3 films firstly grew in an island pattern which then turned into a layer-by-layer pattern when those islands became connected and continuous. Uniform and compact Al2O3 films were obtained when the film thickness reached 40 nm. The average transmittance, optical band gap, and optical absorption coefficient at a wavelength of 1550 nm of this Al2O3 film were about 80%, 3.72 eV, and 3.0 × 104 cm−1, respectively. At a frequency of 1 MHz, the permittivity, dielectric loss, and electrical resistivity were 8.96, 0.31, and 5 × 1010 Ω·cm, respectively. This work provides valuable references for the application of Al2O3 ultrathin films as insulating layers in micro-and opto-electronics.

Probing the mechanical properties of vertically-stacked ultrathin graphene/Al2O3 heterostructures

The superior intrinsic mechanical properties of graphene have been widely studied and utilized to enhance the mechanical properties of various composite materials. However, it is still unclear how heterostructures incorporating graphene behave, and to what extent graphene influences their mechanical response. In this work, a series of graphene/Al2O3 composite films were fabricated via atomic layer deposition of Al2O3 on graphene, and their mechanical behavior was studied using an experimental-computational approach. The inclusion of monolayer chemical vapor deposited graphene between ultrathin Al2O3 films (1.5–4.5 nm thickness) was found to enhance the overall stiffness by as much as 70% compared to a pure Al2O3 film of similar thickness (∼150 GPa to ∼250 GPa). Here, for the first time, the combination of graphene and Al2O3 in vertically-stacked heterostructures results in advanced hybrid films of unprecedented mechanical stiffness that also possess qualities desirable for graphene-based transistors and flexible electronics.

Light-soaking enhanced passivation of Al2O3 on crystalline silicon surface

Negatively charged ultrathin Al2O3 film deposited by atomic layer deposition (ALD) technique has been widely applied in photovoltaic industry because of its excellent passivation effect on both p and n type silicon surfaces. In this study, we have observed an enhancement of the Al2O3 passivation on the surface of p type gallium doped Czochralski silicon under illumination, with a rapid effective lifetime increase by ~90%. However, such passivation effect is not stable in dark circumstance at room temperature, with a slow effective lifetime degradation to the initial value. Furthermore, we have confirmed that the enhancement process under illumination and the degradation process in dark are both thermally activated with activation energies of 0.31 ± 0.02 eV and 0.50 ± 0.07 eV, respectively. The capacitance-voltage (C-V) measurements based on the Al/Al2O3/p-Si capacitor demonstrate that the negative charge at the interface between ultrathin Al2O3 film and p-Si substrate increases from 7.4 × 1011 cm−2 to 9.1 × 1012 cm−2 after 1 h's illumination and then returns to the initial value in dark at room temperature after 3 h. This light induced enhancement effect is wavelength dependent and the cut-off wavelength is in the range of 780–800 nm. Based on these results, we have proposed a model to explain the light-soaking enhanced passivation of Al2O3 on crystalline silicon surface based on a charge trapping and de-trapping process.

Simultaneous Manipulation of O‐Doping and Metal Vacancy in Atomically Thin Zn10In16S34 Nanosheet Arrays toward Improved Photoelectrochemical Performance

The facile hydrothermal synthesis of Zn10 In16 S34 atomically thin nanosheet arrays on fluorine-doped tin oxide glass (FTO) substrates is presented. Through controlling heat treatment in air, O-doping and Zn, S vacancies were simultaneously introduced in Zn10 In16 S34 nanosheets with adjusted phase, morphology, chemical compositions, and energy level distribution. The surface defect states are passivated by depositing ultrathin Al2 O3 film by atomic layer deposition technology. The performance of Zn10 In16 S34 photoanodes is largely improved, with 4.7 times higher current density and reduced onset potential. The experimental results and density functional theory calculations indicate that the enhancement is attributed to the fast photoexcited electron-hole pair separation, decreased surface transfer impedance, prolonged carrier lifetime, and reduced overpotential of oxygen evolution reaction.

Simultaneous Manipulation of O‐Doping and Metal Vacancy in Atomically Thin Zn10In16S34 Nanosheet Arrays toward Improved Photoelectrochemical Performance

AbstractThe facile hydrothermal synthesis of Zn10In16S34 atomically thin nanosheet arrays on fluorine‐doped tin oxide glass (FTO) substrates is presented. Through controlling heat treatment in air, O‐doping and Zn, S vacancies were simultaneously introduced in Zn10In16S34 nanosheets with adjusted phase, morphology, chemical compositions, and energy level distribution. The surface defect states are passivated by depositing ultrathin Al2O3 film by atomic layer deposition technology. The performance of Zn10In16S34 photoanodes is largely improved, with 4.7 times higher current density and reduced onset potential. The experimental results and density functional theory calculations indicate that the enhancement is attributed to the fast photoexcited electron–hole pair separation, decreased surface transfer impedance, prolonged carrier lifetime, and reduced overpotential of oxygen evolution reaction.

Influence of plasma parameters on the properties of ultrathin Al2O3 films prepared by plasma enhanced atomic layer deposition below 100 °C for moisture barrier applications

We report the effect of plasma parameters on the properties of ultrathin Al2O3 films prepared by plasma enhanced atomic layer deposition for moisture barrier applications. The Al2O3 films were grown at 90 °C using trimethylaluminum and O2 plasma as precursors. Plasma power, exposure time and O2 concentration are found to influence the growth behavior, composition and density of ultrathin Al2O3 films. Plasma power ≥ 100 W leads to lower impurity levels and higher mass densities of ∼2.85 g·cm−3. The optimum plasma parameters for our process, a plasma power of 100 W and an exposure time of 3 s, reveal a good water vapor transmission rate of 5 × 10−3 g·m−2·day−1 for polyethylene naphthalate substrates coated with 4 nm-thick Al2O3 films.

A combinatorial approach to enhance barrier properties of thin films on polymers: Seeding and capping of PECVD thin films by PEALD

A combinatorial approach to deposit gas barrier layers (GBLs) on polyethylene terephthalate (PET) by means of plasma‐enhanced chemical vapor deposition (PECVD) and plasma‐enhanced atomic layer deposition (PEALD) is presented. Thin films of SiOx and SiOxCyHz obtained from PECVD were grown either subsequently on a PEALD seeding layer (SiO2) or were capped by ultrathin PEALD films of Al2O3 or SiO2. To study the impact of PEALD layers on the overall GBL performance, PECVD coatings with high macro defect densities and low barrier efficiency with regard to the oxygen transmission rate (OTR) were chosen. PEALD seeding layers demonstrated the ability to influence the subsequent PECVD growth in terms of the lower macro defect density (9 macro‐defects mm−2) and improved barrier performance (OTR = 0.8 cm3 m−2 day−1), while the PEALD capping‐route produced GBLs free of macro‐defects.

Tailoring nanopore formation in atomic layer deposited ultrathin films

Selectivity is a critical attribute of catalysts used in manufacturing of essential and fine chemicals. An excellent way to induce selectivity in catalysts is by using ultrathin films with tailored nanoporosity. For instance, nanopores can be created in atomic layer deposition (ALD) ultrathin over-coatings on supported metal nanoparticles by subjecting the coatings to high temperature annealing. These nanopores expose the active surface of the underlying metal nanoparticles. The dimensions of these nanopores can be tuned to impart shape selectivity: only reactants or products with a specific size or shape can fit inside the pore. In this work, the authors explore the underlying mechanism driving nanopore formation in ALD films. Ultrathin films of ALD TiO2 (∼2.5 nm thick) and ALD Al2O3 (∼4.9 nm thick) were deposited on nonporous γ-Al2O3 nanoparticles. The pore formation and evolution were monitored in situ during thermal annealing using small-angle x-ray scattering (SAXS), and the crystallinity was monitored by in situ x-ray diffraction. A correlation between the nanopore formation and amorphous to crystalline phase transitions in the ALD layers was observed. The authors hypothesize that the pores form through the relaxation of stress induced by densification of the ALD films during the phase transitions. The authors developed a mathematical model to evaluate this hypothesis and found remarkable agreement between the model and the SAXS measurements.